A pullback is a general categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If is a linear transformation between vector spaces, then (usually called transpose map or dual map because its associated matrix is the transpose of ) is an example of a pullback map.
In the case of a diffeomorphism and smooth manifold, a very explicit definition can be formulated. Given an -form on a manifold , define the -form on by its action on an -tuple of tangent vectors as the number . This defines a map on -forms and is the pullback map.